Theorem sqabssub 11020

Description: Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)

Assertion

Ref	Expression
sqabssub	|- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )

Proof of Theorem sqabssub

Step	Hyp	Ref	Expression
1		cjsub 10856	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A - B ) ) = ( ( * ` A ) - ( * ` B ) ) )
2	1	oveq2d 5869	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) x. ( * ` ( A - B ) ) ) = ( ( A - B ) x. ( ( * ` A ) - ( * ` B ) ) ) )
3		cjcl 10812	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
4		cjcl 10812	. . . . 5 |- ( B e. CC -> ( * ` B ) e. CC )
5	3, 4	anim12i 336	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) e. CC /\ ( * ` B ) e. CC ) )
6		mulsub 8320	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( ( * ` A ) e. CC /\ ( * ` B ) e. CC ) ) -> ( ( A - B ) x. ( ( * ` A ) - ( * ` B ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
7	5, 6	mpdan 419	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) x. ( ( * ` A ) - ( * ` B ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
8	2, 7	eqtrd 2203	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) x. ( * ` ( A - B ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
9		subcl 8118	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
10		absvalsq 11017	. . 3 |- ( ( A - B ) e. CC -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( A - B ) x. ( * ` ( A - B ) ) ) )
11	9, 10	syl 14	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( A - B ) x. ( * ` ( A - B ) ) ) )
12		absvalsq 11017	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
13		absvalsq 11017	. . . . 5 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( B x. ( * ` B ) ) )
14		mulcom 7903	. . . . . 6 |- ( ( B e. CC /\ ( * ` B ) e. CC ) -> ( B x. ( * ` B ) ) = ( ( * ` B ) x. B ) )
15	4, 14	mpdan 419	. . . . 5 |- ( B e. CC -> ( B x. ( * ` B ) ) = ( ( * ` B ) x. B ) )
16	13, 15	eqtrd 2203	. . . 4 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( ( * ` B ) x. B ) )
17	12, 16	oveqan12d 5872	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) = ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) )
18		mulcl 7901	. . . . . 6 |- ( ( A e. CC /\ ( * ` B ) e. CC ) -> ( A x. ( * ` B ) ) e. CC )
19	4, 18	sylan2 284	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( * ` B ) ) e. CC )
20	19	addcjd 10921	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) )
21		cjmul 10849	. . . . . . 7 |- ( ( A e. CC /\ ( * ` B ) e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. ( * ` ( * ` B ) ) ) )
22	4, 21	sylan2 284	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. ( * ` ( * ` B ) ) ) )
23		cjcj 10847	. . . . . . . 8 |- ( B e. CC -> ( * ` ( * ` B ) ) = B )
24	23	adantl 275	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( * ` B ) ) = B )
25	24	oveq2d 5869	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) x. ( * ` ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
26	22, 25	eqtrd 2203	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
27	26	oveq2d 5869	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
28	20, 27	eqtr3d 2205	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
29	17, 28	oveq12d 5871	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) = ( ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) - ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) ) )
30	8, 11, 29	3eqtr4d 2213	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A - B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) - ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   ` cfv 5198  (class class class)co 5853   CCcc 7772   + caddc 7777   x. cmul 7779   - cmin 8090   2c2 8929   ^cexp 10475   *ccj 10803   Recre 10804   abscabs 10961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963

This theorem is referenced by:  sqabssubi  11117